Question

A particle is confined to move on the surface of a circular cone with its axis on the \(z\) axis, vertex at the origin (pointing down), and half-angle \(\alpha\). The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates \((\rho, \phi)\) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates ( \(\rho, \phi\) ) and vice versa.

Short answer

The Cartesian coordinates are: \( x = \rho \cos(\phi) \), \( y = \rho \sin(\phi) \), \( z = \frac{\rho}{\tan(\alpha)} \). The generalized coordinates are \( \rho = \sqrt{x^2 + y^2} \), \( \phi = \tan^{-1}(y/x) \).

Step-by-step solution

Understand the Geometrical Setup

The particle is on the surface of a cone with its vertex at the origin and its axis along the z-axis. The cone's half-angle is \(\alpha\). The generalized coordinates given are \( \rho \) (the radial distance from the z-axis) and \( \phi \) (the azimuthal angle around the z-axis).

Express the Height of the Particle

For a cone with half-angle \(\alpha\), any point on the surface of the cone satisfies the relation between \(z\) and \(\rho\):\[ \rho = z \tan(\alpha) \]Therefore, the height \(z\) of the particle is given by:\[ z = \frac{\rho}{\tan(\alpha)} \]

Write Cartesian Coordinates in Terms of Generalized Coordinates

In cylindrical coordinates, the radius \(\rho\) and angle \(\phi\) translate to Cartesian coordinates (x, y) as:- \( x = \rho \cos(\phi) \)- \( y = \rho \sin(\phi) \)The \(z\)-coordinate is expressed in terms of \(\rho\) from Step 2 as:- \( z = \frac{\rho}{\tan(\alpha)} \)Thus, the particle’s Cartesian coordinates are:\[ x = \rho \cos(\phi) \]\[ y = \rho \sin(\phi) \]\[ z = \frac{\rho}{\tan(\alpha)} \]

Express Generalized Coordinates in Terms of Cartesian Coordinates

Given the Cartesian coordinates \(x\), \(y\), and \(z\), the generalized coordinates \(\rho\) and \(\phi\) can be expressed as:- Calculating \(\rho\) as the radial distance in the \(xy\)-plane:\[ \rho = \sqrt{x^2 + y^2} \]- Calculating the azimuthal angle \(\phi\):\[ \phi = \tan^{-1}\left(\frac{y}{x}\right) \]

Key concepts

generalized coordinates

When dealing with complex systems, it can be beneficial to use generalized coordinates. These are coordinates that simplify the mathematical description of the system.
In this scenario, the movement of a particle on the surface of a cone is analyzed using generalized coordinates. Instead of using standard Cartesian coordinates, the problem can be simplified using

• Radial distance (\( \rho \) ) from the z-axis
• Azimuthal angle (\( \phi \) ) around the z-axis.

These coordinates are particularly useful because they naturally follow the cone's shape. They provide a compact way to describe the particle's location with just two parameters, even though the particle is moving in three-dimensional space.
By understanding and using generalized coordinates, we simplify the problem and make it easier to calculate other necessary values.

cylindrical polar coordinates

Cylindrical polar coordinates are a type of generalized coordinate system that expands upon the idea of two-dimensional polar coordinates to three dimensions.
This system consists of

• Radial distance (\( \rho \) ) from the origin to the point
• Azimuthal angle (\( \phi \) ) around an axis (typically the z-axis)
• Height (\( z\) ) along this axis.

In the context of the cone surface, these coordinates effectively describe points by their horizontal and rotational position rather than their linear position in a Cartesian grid.
This is advantageous as it aligns more closely with the natural geometric properties of the cone. For our problem, this system helps by giving a straightforward description of the particle's position on the cone surface, making calculations less complex and more intuitive.

Cartesian coordinates

Cartesian coordinates describe the position of points in space using three values:

• A horizontal distance along the x-axis
• A vertical distance along the y-axis
• A height along the z-axis

They are very familiar and intuitive, often used in basic geometry and algebra to express positions in space.
To convert generalized (cylindrical) coordinates into Cartesian coordinates, we use the following relations:

• \( x = \rho \cos(\phi) \)
• \( y = \rho \sin(\phi) \)
• \( z = \frac{\rho}{\tan(\alpha)} \)

These equations translate positions from cylindrical to Cartesian, allowing us to describe the particle's location in rectangular space.
This is particularly useful when the task requires interactions with systems or equations that are best analyzed in a Cartesian framework.

half-angle of a cone

The half-angle of a cone is a crucial geometric parameter. It is defined as the angle between the cone's side and its axis.
For the exercise at hand, this half-angle is denoted as \(\alpha\). It directly influences the relationship between the radial distance \(\rho\) and the height \(z\) of a point on the cone's surface.
The relation is described by the equation \( \rho = z \tan(\alpha) \). This means that for a fixed angle, increasing \(\rho\) increases \(z\) linearly, following a slope determined by \(\tan(\alpha)\).
Understanding the half-angle helps visualize and predict how the dimensions of the cone will alter as we change the particle's position, making it a vital aspect of the cone dynamics.